Nonsimple material problems addressed by the Lagrange’s identity
 Marin I Marin^{1}Email author,
 Ravi P Agarwal^{2} and
 SR Mahmoud^{3, 4}
https://doi.org/10.1186/168727702013135
© Marin et al.; licensee Springer. 2013
Received: 14 March 2013
Accepted: 4 May 2013
Published: 20 May 2013
Abstract
Our paper is concerned with some basic theorems for nonsimple thermoelastic materials. By using the Lagrange identity, we prove the uniqueness theorem and some continuous dependence theorems without recourse to any energy conservation law, or to any boundedness assumptions on the thermoelastic coefficients. Moreover, we avoid the use of positive definiteness assumptions on the thermoelastic coefficients.
Keywords
1 Introduction
Even classical elasticity does not consider the inner structure, the material response of materials to stimuli depends in a relevant way on its internal structure. Thus, it has been needed to develop some new mathematical models for continuum materials where this kind of effects was taken into account. Some of them are nonsimple elastic solids. It is known that from a mathematical point of view, these materials are characterized by the inclusion of higherorder gradients of displacement in the basic postulates.
The theory of nonsimple elastic materials was first proposed by Toupin in his famous article [1]. Also, among the first studies devoted to this material, we must mention those belonging to Green and Rivlin [2] and Mindlin [3].
The interest to introduce highorder derivatives consists in the fact that the possible configurations of the materials are clarified more and more finely by the values of the successive higher gradients.
As it is known, the constitutive equations of nonsimple elastic solids are known to contain first and secondorder gradients, both contributing to dissipation. It is then interesting to understand the relevance of the two different dissipation mechanisms which can appear in the theory. In fact, the simultaneous presence of both mechanisms can be analyzed as well, with inessential changes in the proofs. In that situation, the behavior turns out to be the same as if only the higherorder dissipation appears in the equations.
In the last decade many studies have been devoted to nonsimple materials. We remember only three of them, differing in issues addressed, though in essence they are dedicated to nonsimple materials. So, in the paper of Pata and Quintanilla [4], the theory is linearized, and a uniqueness result is presented.
Also, the study [5] of Martinez and Quintanilla is devoted to study the incremental problem in the thermoelastic theory of nonsimple elastic materials.
A linearized theory of thermoelasticity for nonsimple materials is derived within the framework of extended thermodynamics in the paper of Ciarletta [6]. The theory is linearized, and a uniqueness result is presented. A Galerkintype solution of the field equations and fundamental solutions for steady vibrations are also studied.
Previous papers on the uniqueness and continuous dependence in elasticity or thermoelasticity were based almost exclusively on the assumptions that the elasticity tensor or thermoelastic coefficients are positive definite (see, for instance, the paper [7]).
In other papers, authors recourse the energy conservation law in order to derive the uniqueness or continuous dependence of solutions. For instance, a uniqueness result was indicated in paper [8] of Green and Lindsay by supplementing the restrictions arising from thermodynamics with certain definiteness assumptions.
We want to outline that there are many papers which employ the various refinements of the Lagrange identity, of which we remember only a few, namely papers [9, 10] and [11]. Also, a lot of papers are dedicated to Cesaro means, as [12–14] and [9] for instance.
The objective of our study is to examine by a new approach the mixed initialboundary value problem in the context of thermoelasticity of nonsimple materials. The approach is developed on the basis of Lagrange identity and its consequences. Therefore, we establish the uniqueness and continuous dependence of solutions with respect to body forces, body couple, generalized external body load and heat supply. We also deduce the continuous dependence of solutions of our problem with respect to initial data and, finally, with respect to thermoelastic coefficients. The results are obtained for bounded regions of the Euclidian threedimensional space. We point out, again, that the results are obtained without recourse to the energy conservation law or to any boundedness assumptions on the thermoelastic coefficients. Also, we avoid the use of definiteness assumptions on the thermoelastic coefficients.
2 Basic equations
We assume that a bounded region B of the threedimensional Euclidian space ${R}^{3}$ is occupied by a nonsimple elastic body, referred to the reference configuration and a fixed system of rectangular Cartesian axes. Let $\overline{B}$ denote the closure of B and call ∂B the boundary of the domain B. We consider ∂B to be a piecewise smooth surface and designate by ${n}_{i}$ the components of the outward unit normal to the surface ∂B. Letters in boldface stand for vector fields. We use the notation ${v}_{i}$ to designate the components of the vector v in the underlying rectangular Cartesian coordinates frame. Superposed dots stand for the material time derivative. We employ the usual summation and differentiation conventions: the subscripts are understood to range over integer $(1,2,3)$. Summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate.
The spatial argument and the time argument of a function will be omitted when there is no likelihood of confusion. We refer the motion of the body to a fixed system of rectangular Cartesian axes $O{x}_{i}$, $i=1,2,3$. Let us denote by ${u}_{i}$ the components of the displacement vector and by θ the temperature measured from the constant absolute temperature ${\theta}_{0}$ of the body in its reference state.
As usual, we denote by ${\sigma}_{ij}$ the components of the stress tensor and by ${m}_{ijk}$ the components of the hyperstress tensor over B.
We here will use the theory and the notation in the way developed by Iesan in his book, which tackles also nonsimple materials [15].

${f}_{i}$ the components of body force;

ϱ is the reference constant mass density;

${\sigma}_{ij}$ and ${m}_{ijk}$ are the components of the stress;

S is the entropy per unit mass;

r is the heat supply per unit mass;

${q}_{i}$ are the components of heat flux vector.
at regular points of the surface ∂B. Here, ${n}_{i}$ are the components of the outward unit normal of the surface ∂B.
where ${t}_{0}$ is some instant that may be infinite.
Assume that ${c}_{i}$, ${d}_{i}$, σ, ${\overline{u}}_{i}$, ${\overline{t}}_{i}$, $\overline{\theta}$ and $\overline{q}$ are prescribed smooth functions in their domains.
 (i)
all constitutive coefficients are continuously differentiable functions on $\overline{B}$;
 (ii)
ϱ is continuous on $\overline{B}$;
 (iii)
${f}_{i}$ and r are continuous functions on $\overline{B}\times [0,{t}_{0})$;
 (iv)
${c}_{i}$, ${d}_{i}$ and σ are continuous functions on $\overline{B}$;
 (v)
${\overline{u}}_{i}$ and $\overline{\theta}$ are continuous functions on $\partial {B}_{1}\times [0,{t}_{0})$ and $\partial {B}_{2}\times [0,{t}_{0})$, respectively;
 (vi)
${\overline{t}}_{i}$ and $\overline{q}$ are piecewise regular functions on $\partial {B}_{1}^{c}\times [0,{t}_{0})$ and $\partial {B}_{2}^{c}\times [0,{t}_{0})$, respectively, and continuous in time.
By a solution of the mixed initial boundary value problem of the theory of thermoelasticity of nonsimple bodies in the cylinder ${\mathrm{\Omega}}_{0}=B\times [0,{t}_{0})$, we mean an ordered array $({u}_{i},\theta )$ which satisfies the system of equations (8) for all $(x,t)\in {\mathrm{\Omega}}_{0}$, the boundary conditions (7) and the initial conditions (6).
3 Main result
For the sake of simplicity, the spatial argument and the time argument of the functions $f(t,x)$ and $g(t,x)$ are omitted because there is no likelihood of confusion.
respectively.

the equations of motion$\begin{array}{rl}\varrho {\ddot{v}}_{i}=& {A}_{ijrs}{v}_{r,sj}+{B}_{ijpqr}{v}_{r,pqj}+{E}_{ij}{\chi}_{,j}\\ +{B}_{mnsji}{v}_{m,nsj}+{C}_{sjimnr}{v}_{r,mnsj}+{D}_{sji}{\chi}_{,sj}+\varrho {\mathcal{F}}_{i};\end{array}$(12)

the equations of energy$c\dot{\chi}+{\theta}_{0}({E}_{ij}{\dot{v}}_{i,j}+{D}_{ijk}{\dot{v}}_{k,ij})={({k}_{ij}{\chi}_{,j})}_{,i}+\varrho \mathcal{P};$(13)

the initial conditions${v}_{i}(x,0)=0,\phantom{\rule{2em}{0ex}}{\dot{v}}_{i}(x,0)=0,\phantom{\rule{2em}{0ex}}\chi (x,0)=0,\phantom{\rule{1em}{0ex}}x\in \overline{B};$(14)

the boundary conditions$\begin{array}{r}{v}_{i}(x,t)=0\phantom{\rule{1em}{0ex}}\text{on}\partial {B}_{1}\times [0,{t}_{0}),\phantom{\rule{2em}{0ex}}{t}_{ji}(x,t){n}_{j}=0\phantom{\rule{1em}{0ex}}\text{on}\partial {B}_{1}^{c}\times [0,{t}_{0}),\\ \chi (x,t)=0\phantom{\rule{1em}{0ex}}\text{on}\partial {B}_{2}\times [0,{t}_{0}),\phantom{\rule{2em}{0ex}}{p}_{i}(x,t){n}_{i}=0\phantom{\rule{1em}{0ex}}\text{on}\partial {B}_{2}^{c}\times [0,{t}_{0}).\end{array}$(15)
We are now in a position to prove the first basic result.
where we have used the fact that the initial and boundary data are null.
We shall eliminate the inertial terms on the righthand side of the relation (17) by means of the equations of motion for the differences $({v}_{i},\chi )$.
We now substitute (25) in (22), and so we are led to equality (16). With this, the proof of Theorem 1 is completed. □
Remark It is important to note that the identity (16) is just like in the classical thermoelasticity (see [17]).
The identity (16) constitutes the basis on which we shall prove the uniqueness and the continuous dependence results.
We proceed first to obtain the uniqueness of the solution of the mixed initial boundary value problem defined by (8), (6) and (7).
Also, we suppose that the symmetry relations (5) are satisfied. If $\partial {B}_{2}$ is not empty or $c(x)\ne 0$ on B, then the mixed initial boundary value problem in thermoelastodynamics of nonsimple materials has at most one solution.
Proof Suppose, by contrary, that our mixed problem defined by (8), (6) and (7) has two solutions $({u}_{i}^{(\alpha )},{\theta}^{(\alpha )})$ ($\alpha =1,2$) that correspond to the same initial and boundary data, to the same body force and the same heat supply.
If $\partial {B}_{2}$ is not empty, considering the boundary conditions (7), then from (28) we deduce that (27) holds. If $a(x)\ne 0$, from the equation of energy (written for the differences), we get $\dot{\chi}=0$. However, χ vanishes initially, such that (27) again holds true.
and repeat the above procedure on the interval $[{t}_{0}/2,{t}_{0}/2+{t}_{0}/4]$ such that we extend the conclusion (27) on $B\times [0,3{t}_{0}/4)$, and so on.
Finally, we obtain (27) on $B\times [0,{t}_{0})$ and this concludes the proof of Theorem 2. □
We are ready to state and prove the continuous dependence theorem with regard to body force and heat supply on the compact subintervals of the interval $[0,{t}_{0})$ for the solution of the mixed initial boundary value problem defined by the system of equations (8), the initial conditions (6) and the boundary conditions (7).
where ${v}_{i}(s)$ and $\chi (s)$ are the differences defined in (26) and $s\in [0,{t}_{\ast}/2)$.
Proof We will use the identity (16). On the righthand side of this identity, we employ the Schwarz inequality for each integral.
where, at last, we use the substitution $2ts\to s$.
We proceed analogously with other integrals in the identity (16). Finally, we integrate the resulting inequality over $[0,s]$, $s\in [0,{t}_{\ast}/2]$ and we obtain the inequality (30) and the proof of Theorem 3 is complete. □
In the following theorem, we use the estimate (30) in order to deduce a continuous dependence result upon initial data.
By using these specifications, the estimate (32) follows from (30) and Theorem 4 is concluded. □
Finally, we obtain a continuous dependence result of the solution to problems (8), (6) and (7) upon the thermoelastic coefficients, again as a consequence of Theorem 3.
Thus the problem is analogous to the problem from Theorem 4. Therefore, according to the estimates (32) and (30), we obtain the desired result. □
4 Concluding remarks
The uniqueness theorem and the continuous dependence theorems were proved without recourse to any conservation laws or to any boundedness assumptions on the thermoelastic coefficients. In various papers, the existence of the solution to the mixed initial boundary value problem defined by (8), (6) and (7) is obtained by assuming some strong restrictions. For instance, in the paper [18] the existence of the solution is obtained under assumption that the internal energy density is positive definite.
Declarations
Acknowledgements
We express our gratitude to the referees for their valuable criticisms of the manuscript and for helpful suggestions.
Authors’ Affiliations
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